Intuition for the CRVTW uplift model
Joseph Bolton
22 April, 2020
...................................................................... ...................................................................... ..............................$$$..................................... ...................................$$$$............................... .................................$$$$$$$.............................. .................................$$$$$$$$............................. ..................................$$$$$$.............................. .........................$$$$$........................................ ........................$$$$$$$....................................... .........................$$$$$$.....$$$............................... .......................................$$$............................ ......................................$$$$$........................... .................................$$$..$$$$$........................... ...................................................................... ............................$$$$$....$................................ ............................$$$$...................................... ...................................................................... ..........................$$$$$$$$$$$$$$$$............................ ........................$$................$$.......................... ........................$$.................$$......................... .........................$$..............$$$.......................... ..........................$...........$..$$........................... ..........................$...........$..$$........................... ..........................$..............$$........................... ..........................$..............$$........................... ..........................$..............$$........................... ..........................$.....$$$$.....$$........................... ..........................$.....$$$$$....$$........................... ..........................$......$$......$$........................... ..........................$..............$$........................... ..........................$..............$$........................... .........................$$...............$$.......................... ........................$............$.....$$......................... ......................$$...........$$$$$....$$........................ .....................$$.............$$$$......$$...................... ....................$$.........................$$..................... ..................$$............................$$.................... .................$$..$$$$$$$$$$..$$$$$$$.....$$$..$$.................. ................$$..$$$$$$$$$$$.$$$$$$$$......$$$..$$................. ..............$$..$$$$$$$$$$$$$$$$$$$$$$......$$$$...$................ .............$$..$$$$$$$$$$$$$$$...$$$$$$...$$$$$$$$..$$.............. ............$...$$$$$$$$$$$$$$$$$.$$$$$$$$$$$$$$$$$$$..$$............. ..........$$..$$$$$$$$$$$$$$$$$$$$$$$....$$$$$$$$$$$$$...$$........... .........$$..$$$$$$$$$$$$$$$$$$...$$$....$$$$$$$$$$$$$$$..$$.......... ........$$..$$$$$$$$$$$$$$$$$$$..$$$$$$$$$$$$$$$$$$$$$$$$..$$......... ......$$...$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$...$$....... .....$$..$$$$$$$$$$$$$$$$$$$$$$$$..$$$$$$$$$$$$$$$$$$$$$$$$$..$$...... ....$$..$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$..$$..... ....$..$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$..$..... ....$..$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$..$..... .....$.........................................................$$..... ......$$$...................................................$$$....... ..........$.$..........................................$$$$........... ......................................................................
This page is a very brief intuitive introduction to the Continuous Response Variable Transformation with Weightings (CRVTW) uplift model discussed by the paper:
“Response Transformation and Profit Decomposition for Revenue Uplift Modeling”
[Robin M. Gubela, Stefan Lessmann, Szymon Jaroszewicz. 2019]
Consider the following data, in which we are interested in estimating how an administered treatment has affected outcome \(y\) for units with different values of feature \(x\).
There appears to be a treatment effect somewhere between \(40<x<60\).
We create the following transformed outcome variable:
\[z_i \quad=\quad \begin{cases} \hspace{3.5mm}\Bigg(\displaystyle\frac{n}{n_{treated}}\Bigg) \cdot y_i \quad \text{ if unit } i \text{ received treatment} \\ - \Bigg(\displaystyle\frac{n}{n_{control}}\Bigg) \cdot y_i \quad \text{ if unit } i \text{ did not receive treatment (control group)} \end{cases}\]
Here is how this transformed outcome variable \(z\) looks:
We fit a natural spline regression model to the transformed outcome \(z\):
This fitted model gives an estimate of the Conditional Average Treatment Effect (CATE) \(\space\tau(x)\) which, in a random sampling context, is:
\[\tau(x) \quad=\quad E\Big[Y_i\space\Bigl|X_i=x, \space\text{Treatment}=Yes\Big] \quad\mathbf{-}\quad E\Big[Y_i\space\Bigl|X_i=x, \space\text{Treatment}=No\Big]\]
It does this since:
\[E\Big[Z_i\Bigl|X_i=x\Big] \quad=\quad E\Big[Y_i\space\Bigl|X_i=x, \space\text{Treatment}=Yes\Big] \quad\mathbf{-}\quad E\Big[Y_i\space\Bigl|X_i=x, \space\text{Treatment}=No\Big]\]
Here is a comparison of the estimated uplift effect with the true uplift effect (I only know the true uplift effect because I simulated it):
